This page provides several examples on how to calculate and estimate the diameter for a guqin string, specifically multifilament core guqin strings. This method is primarily useful for determining calculations for silk strings in particular, however, it is also extremely useful for calculating the diameters of twisted core strings using other materials, such as synthetics or even metal, and can be used for any material. I have used this method extensively in my own string making endeavors, and have found it to be one of the key and crucial starting points for beginning to design strings. This will help you determine the total thread count, substrand count, and resulting string diameter, regardless of material. While it is not absolutely 100% accurate, as there are some trial and error discrepancies involved, it is still nevertheless a very accurate and powerful tool.
Essentially, multifilament twisted core musical strings, particularly those made of organic materials such as silk or gut, are really just specialized ropes optimized for musical note production. They are essentially just a bunch of tiny fibers twisted together in larger bundles, stretched, glued, and dried under various conditions and manufacturing methods. However, the basic rope making method which I currently utilize for my own experimental synthetic twisted core guqin strings also use these methods, and indeed as mentioned above, musical strings of this nature have been most likely derived from rope making technology. As such, calculations and estimates can be made for a particular string diameter regardless of material when utilizing this method of manufacture, and can be further refined and advanced by accounting for material deformation and elasticity. To figure out how to find the total diameter of a string, first for silk, and to further correlate this to other synthetics such as nylon, polyester, or Kevlar threads, as well as thin gauge metal strands, calculations are made using circle-in-circle calculation estimates to work from the known starting parameters, such as fiber diameter, to a final string diameter. These are just the basic geometric estimates, and do not include more advanced compensation factors such as material deformation and elasticity – however, this simple method alone is powerful enough to obtain accurate, repeatable results, as well as provide a solid starting point in making such strings. The calculator I refer to for the calculations below can be found at The Engineering ToolBox website, located at this link: http://www.engineeringtoolbox.com/smaller-circles-in-larger-circle-d_1849.html. For each calculation and step, I will provide screenshots to make it easier to follow and visualize the process.
I will break this example calculation guide into two parts. The first part will deal with calculating silk string diameters, using examples from the two major methods of silk string construction – full twisted bundles, as seen in Suxin silk strings, and rope twisted strings using larger twisted substrands, as seen in most other silk guqin strings, and other silk strings, such as those for the shamisen. As you will see, there is a radical difference in the resulting string between these two methods, and should be considered carefully when designing your own strings. The second part deals with an example calculation for a rope twisted synthetic string made from nylon thread, and illustrates how to match and correlate the design of this material to that of an equivalent silk string. As far as I am aware, I have not yet come across any other site or resource that goes into the actual calculation details of string diameter estimation for silk strings or similarly related synthetic rope-twisted strings for the guqin, and perhaps for other instruments that may benefit from these types of strings as well.
PART I – SILK STRINGS
Individual silk fibers from the bombyx mori silk worm are about 13um, or microns, in diameter. To put this in perspective, the average human hair is about 181um in diameter – these are some really tiny fibers! However, it takes many of these fibers to create the substrands of a string, which are further twisted together to make a full string. It should be noted that silk fibers are also not uniform in cross-section – they are actually more like rounded triangles, so these calculations will provide only an estimate for a desired string diameter using silk, as for ease of calculation, it assumes the fibers are uniformly circular in cross-section. Nevertheless, this provides a very powerful tool for string making, and makes starting the design of strings much easier. As you progress in experience, adjustments and refinements will naturally be included in the process to compensate.
On John Thompson’s website, there is a wealth of information on silk strings, the best resource on the internet that I have found so far on silk strings for the qin. One page, found here, http://www.silkqin.com/02qnpu/05tydq/ty1b.htm#strands, has a very useful table on collected information for the diameters of many types of silk strings. These diameters will provide the first basis of parameters for our calculation estimates. Silk filaments are extremely thin, so they are bundled together in larger numbers to make a silk thread, to be further twisted with similar threads. These numbers can vary, and from John Thompson’s site it is cited that this could range on average from 9 to 12 filaments per thread. Let us pick the 7th string, standard size, as an example for calculation estimates. You can find the table on the page in the link above. This diameter correlates to 0.85mm. Let us also pick a construction technique that just uses a single twisted bunch of threads which are each made from a particular number of filaments. Assuming an average diameter of 13um for a single silk filament, and uniform cross-section (silk has kind of a rounded-triangular cross-section, but for simplicity and rough estimate we go with round), this comes out to 0.013mm. Now we need to choose how many filaments to estimate per thread. If we choose 9 filaments, then using the circle-in-circle calculator linked in the introduction above, we plug 0.013 into the outside diameters of inside smaller circles, and plug an arbitrary number into the box above, labeled inside diameter of outside larger circle box, say 0.1, to get the number of filaments per thread – in this case, it comes out to 43. See Figure 1 below:
Our goal is 9, so we decrease the diameter of the larger circle until we get to 9. In this case, we get about 0.047mm diameter for 9 filaments (Figure 2), and for the other estimate of 12 filaments, we get 0.0541mm diameter (Figure 3). Note the inner diameter of the outer larger circle is not exactly perfectly tight to all of the outer diameters of the inner circles – for estimates, or plugging in known sizes of fibers, this is fine. When dealing with optimizing a string and going through trial and error calculations to get the exact size, you will want to get these circle perimeters touching as close as possible.
Now we go to the calculator again, except this time we plug in 0.85 into the inside diameter of outside larger circle box, and 0.047 or 0.0541 into the outside diameters of inside smaller circles box to get the total number of threads. For the 9 filament, 0.047mm diameter thread, we get a total of 251 threads (Figure 4), and for the 12 filament, 0.0541mm diameter thread, we get 189 total threads (Figure 5).
For the filament count using this method, we get a total of 2259 filaments for a 9 filament per thread, and 2268 total filaments for 12 filaments per thread. These numbers should be equal, but are very slightly different due to the discrepancy in any residual excess spacing between substrands in the calculator. Keep this number in mind though, for you will notice this will be very different from the second construction method described below, which has massive implications for string response, properties, and ultimately timbre. This method of construction, as mentioned above, creates a string where all of the threads are twisted together as a single bundle. This results in a smoother string, but is much less commonly seen for guqin strings nowadays. Suxin strings use this method, and this is actually very similar to the method used for making traditional gut strings for instruments such as the violin.
The other way to make silk strings is to utilize rope-making methods, in which one first creates larger substrands, twisting these together first in one direction from the thin threads, and then twisting these substrands together in the opposite direction to make the final string. This is more along the lines of a “true” rope structure. Generally, you will find that 3 or 4 substrands is a common number. I have found that silk qin and shamisen strings, made from this method, are most often made with 4. Note that this method will result in a decrease in the total number of threads, which can have large implications on the response and properties of the string, as described shortly below. In order to get the total number of silk threads, we must back-calculate from total diameter, to substrand diameter, to thread number. Let us take 4 substrands for this example of a 0.85mm diameter silk 7th qin string. In the calculator, we first put 0.85 into the top box, and some arbitrary number into the bottom box, say 0.4, which gives us a string diameter slightly larger than that with a total of 2 substrands. See Figure 6 below:
We need 4, so we increase the number in the second box of the calculator until we reach 4. This ends up being about 0.352, which is our diameter per substring, in mm. Note that this diameter should be exact, unlike the previous above figure, which was just a starting estimate. See Figure 7 below:
We then plug this number into the top box, and our thread size into the bottom box. For a 9 filament thread of 0.047mm diameter, we get 41 threads (Figure 8), and for a 12 filament thread of 0.0541mm diameter, we get 31 threads (Figure 9).
So for our silk estimate for an average standard silk 7th qin string, we have a range of filament numbers to choose from, based on silk thread size. For a string that uses thread substrands made from 9 filaments each, with 41 filaments per substrand, and 4 substrands total, we get a total of 1476 filaments, and for a string that uses thread substrands made from 12 filaments each, with 31 filaments per substrand, and 4 substrands total, we get a total of 1488 filaments. Note the similarly very small discrepancy between these two numbers like in the estimate above due to spacing discrepancy in the calculator. Also note however that for an equivalent diameter string, this method results in a bit more than half the total number of silk filaments used over the first method. This difference between two construction techniques for the same string, assuming the same glues are used for each results in one string NEARLY TWICE THE DENSITY OF THE OTHER, despite being the same diameter, material, and thread size! However, other factors, such as internal damping and flexibility will be drastically altered as well, which all end up affecting the overall timbre of the string. Also note that since more silk material is being used for the first method than the second, the overall cost of the string will increase as well.
PART II – SYNTHETIC STRINGS
We can now apply the same principles to a synthetic thread. I have successfully used and implemented nylon, polyester, and Kevlar threads for qin strings, using these calculations as the starting point for determining how many threads to use for a given string diameter. For these strings, I choose a thread with as small of a diameter as I can find. So far, this has been about Tex 16, or Size 15 for all of these materials. The thread diameters are similar between the materials at this size, but still vary slightly. For this example, I will use the nylon thread that I have used for my experimental twisted core nylon strings, with a diameter of about 0.0048″, or about 0.122mm. Note that in comparison with our silk thread estimates, using the circle-in-circle calculator, we find that a 0.122mm nylon thread will be slightly larger than the size of 4 bundled 0.047mm silk threads with 9 filaments each (Figure 10) or slightly larger than 3 bundled 0.0541mm silk threads with 12 filaments each (Figure 11).
Again these are not exact figures, but rough estimates for use as a guideline in decision making for strings. With a total string diameter of 0.85mm for a standard size 7th silk thread, plugging this number and 0.122 into the calculator, we get about 35 total threads per string. See Figure 12 below:
However, this is NOT our correct answer. Since I am not making these strings with glue, but rather as twisted ropes without glue, I need to first make larger substrands to twist into the final string. For this number, you can generally choose either 3 or 4 substrands. I found that for the material I am using, 3 works the best and results in a much smoother strings, and is all around easier to work with. The number of threads you use per substrand is then found out through a bit of trial and error and optimization, by just plugging in numbers and refining the result until you reach your desired outcome. It is easier in this case if we backtrack from final string, to substrands, to threads. So in our calculator, we plug in 0.85 in the top box, and some random number into the bottom. Let’s just say 0.1, which results in a total of 54 substrands – way more than 3. See Figure 13 below:
We keep lowering this number until we reach 3 – for this example, it turns out to be 0.394mm. See Figure 14 below:
We then plug this number into the top box, and our thread diameter of .122 into the bottom box. This gives us a total thread count of about 7 threads per substrand, for a total of 21 threads per string, divided into 3 substrands. Note that the resulting diameter is slightly larger than the total of 7 threads for this particular substrand. See Figure 15 below:
Yet we are still not done here, and this again is NOT our final answer. Based on the rope making method I employ, there will always be an even number of threads per substrand, since you wind the string between the two hooks for one complete pass, and to tie them together at the same end, this results in 2 threads per pass. Therefore, we need to choose to round this number, in this case 7, either up or down. Remember earlier that I talked about density – since nylon is less dense than silk, it would require an equivalently slightly thicker string. Therefore, rounding the number of threads per substrand up to 8, I now get a total diameter of 0.405mm per substrand (Figure 16), and with 3 substrands per string, I get a total string diameter of about 0.875mm (Figure 17) – low and behold, this falls very slightly above our original silk 7th string diameter of 0.85mm!
So, our first estimate for our average standard size 7th string for the guqin made of nylon with a thread diameter of 0.122mm, with 8 threads per substrand, and 3 substrands total, we get a string diameter of 0.875mm. Our back calculations and estimates falls right exactly in the range we want when comparing to its similar silk string counterpart of 0.85mm! This example turns out to be the exact numbers I used for my optimized nylon 7th string, which is noted as Trial #13 – the twisting parameters as well as harmonic analysis data for this exact string can be found here in the Nylon Guqin String Trial #13 page.
Something else to be mindful of is the total strength of the string. A rope will be strongest with parallel laid lines – so the theoretical max strength of our string BEFORE twisting should be the strength per thread multiplied by the number of total threads. 3 substrands of 8 threads each gives us 24. The nylon I am using has a strength of about 2lbs per thread, so 2lbs x 24 threads = 48lbs total. However, twisting the strands together will weaken this structure, so use this figure very conservatively in your calculations, and plan for enough overhead in strength that your string can handle the tension of stringing. For both nylon and polyester for the strings I have made, I have not had a single issue in breakage yet, even tuning the thinnest 7th string up to modern standard tuning of D3. Also note that if you are making string that involves some sort of cooking or gluing process, diameters will change based on glues and absorption of moisture into the material. The above estimates are mainly for determining the rough goal diameter of a particular string using dry twisted methods, and though they are rough theoretical estimates, they provide very valuable insight, as well as an excellent and reliable method for estimating string diameters when experimenting with constructing multifilament twisted cores, using any material. Also of important note is that even with using the same thread size, different materials will have different properties of stretch, compression, and packing characteristics that will make the resulting strings slightly different in diameter. For example, I have found that for equivalent construction parameters and filament number strings between nylon and polyester, the resulting polyester strings are always thinner than their nylon counterparts.